The author proves that the surjectivity of a partial differential operator \(P(D)\) in the space \({\mathcal E}_{\{M_ \alpha\}}(\Omega)\) of ultradifferentiable functions is equivalent to the same property in the space \({\mathcal D}_{(M_ \alpha)}(\Omega)\) of Beurling ultradistributions, and can be characterized in terms of the \(P\)- convexity of the open domain \(\Omega\). When \(M_ \alpha=e^{c\alpha^ 2}\), \(c>0\), the general result implies that \(P(D)\) is surjective in \({\mathcal E}_{\{M_ \alpha\}}(\Omega)\) if and only if it is surjective in \({\mathcal D}'(\Omega)\).